On birefringence phenomena, associated with total reflection B. Julia, A. Neveu To cite this version: B. Julia, A. Neveu. On birefringence phenomena, associated with total reflection. Journal de Physique, 1973, 34 (5-6), pp.335-340. 10.1051/jphys:01973003405-6033500. jpa-00207388 HAL Id: jpa-00207388 https://hal.archives-ouvertes.fr/jpa-00207388 Submitted on 1 Jan 1973 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Tome 34 N° 5-6 MAI-JUIN 1973 LE JOURNAL DE PHYSIQUE Classification Physics Abstracts 08.10 - 08.20 - 18.10 ON BIREFRINGENCE PHENOMENA, ASSOCIATED WITH TOTAL REFLECTION B. JULIA and A. NEVEU Laboratoire de Physique Théorique et Hautes Energies, Orsay (*) (Reçu le 10 novembre 1972) Résumé. 2014 Nous étudions les propriétés de biréfringence de la réflexion totale d’un faisceau lumineux monochromatique, sur le plan de séparation de deux milieux homogènes isotropes, par des méthodes de déphasage. Ceci explique à la fois l’effet Goos-Hänchen longitudinal dans lequel une source de lumière monochromatique non polarisée donne deux images de polarisations rectilignes orthogonales, et l’effet transverse étudié récemment par Imbert, pour lequel les deux images sont polarisées circulairement. Nous donnons une méthode simple pour déterminer les polarisations et les positions des images d’une source cylindrique. Abstract. 2014 We study by phase-shift methods the birefringence properties of total reflection of a light beam at the separation plane between two isotropic homogeneous media. This includes the Goos-Hänchen longitudinal effect in wich a source of unpolarized light is split into two images, each of them linearly polarized, as well as the transverse shift recently investigated experimentally by Imbert, where the effect is between left and right circular polarizations. We give a general simple method for determining the polarizations and positions of the images. Introduction. - The existence of birefringence pro- unphysical point in the computation of the various perties of total reflection of light on the separation effects with the Poynting vector is the use of plane plane of two isotropic homogeneous media has been waves of a finite width, neglecting what may occur known for some time [1 ], and experimental evidence near the edges. For these reasons, an approach using was first demonstrated by Goos and Hanchez [2]. phase-shifts seems safer to us, in particular because These authors considered a large number of successive it is more closely related both to the exact solution of total reflections of a pencil of light between two Maxwell’s equations, and to the mechanism of for- parallel planes, and found that a source of unpola- mation of images in geometrical optics. rized light is split into two images, one image being In this paper, we show that one can compute all polarized parallel to the incidence plane, the other effects which have been experimentally observed by perpendicular. Using a different apparatus, Imbert [3] using the exact solutions of Maxwell’s equations for has recently found evidence for a splitting between the reflection of polarized plane waves on the sepa- right and left circular polarizations (see also [10]) : in ration plane of two homogeneous isotropic media, his experiment, the successive total reflections of the together with some simple geometrical optics approxi- pencil of unpolarized light take place on the sides of a mations, which turn out to be always valid in expe- regular dielectric prism. rimental situations. We deal with the usual longitu- These effects, which are of the order of one wave- dinal Goos-Hânchen effect in section 1, and with length at each reflection, have been computed by Imbert’s transverse shift in sections 2 and 3. It turns out various methods [4], [6], [7] among which [4] the use that the geometrical arrangement of the experimental of Poynting vectors is very popular. In our opinion apparatus is of crucial importance in the determination however, if the Poynting vector may be useful to of the polarizations of the two images of an un pola- obtain an order of magnitude of the effect, it cannot rized source. provide a completely consistent treatment of the problem : in particular because it does not tell clearly 1. Longitudinal shift (Goos-Hânchen). - The lon- which polarization states are relevant ; another gitudinal shift is the easiest to compute, owing to the simpler geometrical structure of the system : we consider (Fig. 1) a rectilinear source of light S (narrow (*) Laboratoire associé au Centre National de la Recherche slit for instance), emitting a pencil of of small Scientifique. Postal address : Laboratoire de Physique Théorique light This is reflected on the et Hautes Energies, Bâtiment 211, Université de Paris-Sud, aperture Brn. pencil totally 91405 Orsay, France. plane surface between the vacuum and an isotropic Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01973003405-6033500 336 medium with index n. This separation plane is parallel to the source ; Oy is its intersection with the plane in which figure 1 is drawn. Let i be the mean angle of incidence of the narrow pencil emitted by S. We can decompose this pencil into a linear superposition of plane waves, which have slightly different incidence angles on Oy. Each plane wave can then be reflected on the surface Oy, accord- ing to Maxwell’s theory. We then consider the set of the reflected plane waves. In the approximation of geometrical optics (to be justified later), these reflected waves recombine by superposition to a pencil which, in lowest order in Brn, comes from the virtual cylindrical source 1 : 1 is just the center of curvature of the envelope of the reflected plane waves. 1 will in general be different from S’, the symmetric of S with respect to Oy, and its position will depend on the polarization, thus giving rise to the effect. FIG. 1. - Goos-Hânchen Reflection It is straight forward to compute the location of I : coordinates in the S’ its (S’ X, Y) system (see Fig. 1) a finite portion of the caustic surface, the separation are : of polarizations is complete only if the distance between the two images is larger than their spreading out : If this condition is not satisfied, one can still see the where À is the wavelength of the light in vacuo, and 5(i) Goos-Hânchen effect by using light polarized, perpen- the phase shift of a plane wave at total reflection, as dicular or parallel, to the incidence plane. computed from Maxwell’s theory. Going to the expres- Let us now briefly discuss the validity of the approxi- sion of 5 in terms of i and n for polarizations perpen- mation of geometrical optics used in our derivation dicular and parallel to the incidence plane [5], one of formulae (1)-(4). Indeed, it is in principle unsafe finds the explicit expressions : to use geometrical optics to obtain a result like (1)-(4) which is of the order of a wavelength, that is to say of the same order of magnitude as diffraction. However, in the experimental apparatus, the beam is reflected N times between two reflection planes like Oy, in order to amplify the effect. In this process, it is clear that indeed phase-shifts add up, whereas the magnitude of diffraction, which is set by the aperture of the beam, remains constant. Thanks to this multiplying effect, the two images are split by N(L11.L - 11 111), which can be made much larger than diffraction. Hence, whereas it would be meaningless for a single reflection, our of B"1’) approximation geometrical optics is justified for the full experiment. As i varies, I1 and III define two caustic surfaces Transverse shift and which are the images of S through our anisotropic 2. approximate analysis of optical system. Imbert’s experiment. - In [3], Imbert describes a nice Several remarks are in order : formulae (2) and (4) experiment which shows that total reflection can also do not seem to exist in the literature on the subject. give rise to a splitting between the two circular pola- Our results for the experimentally observed quan- risations. He also computes this effect by using the tities L1 LL and All1 agree with those of Hora [6] and Poynting vector of the evanescent wave. What we Boulware [7] who also derives them via a phase-shift want to show in this section is that the effect can be analysis. However, they differ by a factor cos2 i/cos2 ij, computed by using phases, just as the longitudinal with sin il = 1/n from the formula of Renard [8] and effect of section 1. The origin of this new effect on Imbert who, however, derive them using a heuristic circular polarizations is more subtle than the effect of [3] ° argument with Poynting vectors. section 1, and rests essentially on the rather complex Since the image of a beam with a finite aperture Brn is geometrical structure of Imbert’s experiment. 337 In this section, we shall only give an approximate arrangement is such that the light follows a helical treatment of the effect, leaving the complete analysis path in the prism. The view of that path from under is for the next section. Indeed, we shall now show that drawn on figure 3.